## A Mathematician's Lament — an Indictment of US Math Education 677

Scott Aaronson recently had "A Mathematician's Lament" [PDF], Paul Lockhardt's indictment of K-12 math education in the US, pointed out to him and takes some time to examine the finer points.

*"Lockhardt says pretty much everything I've wanted to say about this subject since the age of twelve, and does so with the thunderous rage of an Old Testament prophet. If you like math, and more so if you think you don't like math, I implore you to read his essay with every atom of my being. Which is not to say I don't have a few quibbles [...] In the end, Lockhardt's lament is subversive, angry, and radical ... but if you know anything about math and anything about K-12 'education' (at least in the United States), I defy you to read and find a single sentence that isn't permeated, suffused, soaked, and encrusted with truth."*
## Can't count (Score:5, Funny)

second!

## Re: (Score:3, Funny)

But he got to base 2...

## Re: (Score:3, Funny)

Because all your *other* base are belong to US!

## Several Proxies (Score:5, Informative)

From what I can tell, they all look to be the same length and size and hopefully are not older revisions of this paper.

## Re:Several Proxies (Score:5, Funny)

For example (FTS):

And just how, pray tell, are we supposed to read his essay with every atom of your being?

I mean sure, I could read his essay with every atom of

mybeing, but wouldn't it violate some mathematical and physical principles for me to read it with the submitter's being?## Re: (Score:3, Insightful)

Next time link to a video of someone getting a baseball in the nuts, they'll love that..

## Re:Several Proxies (Score:4, Interesting)

Bah, like we're going to RTFA on a Friday when there are much better ...

I know you're mostly joking but this was a pretty interesting albeit lengthy opinion piece. In fact, he even busts into dialogue between two fictional characters named Simplicio & Salviati to illustrate his point. It's a very Plato/Caroll/Hofstadter sort of way to illustrate his point. Hell, I love this format so much, half my posts are in it [slashdot.org]!

Anyway, after reading this, I am really eager for vdash.org [slashdot.org] to get its wiki up and running so that can be used to build engines and homework for students. Maybe even provide a hub for teachers to discuss interesting assignments? I'm sure the discussion pages will prove interesting if real academics get in arguments about proofs and math. I don't think the real payoff would be reinstitutionalizing the teachers but instead giving the students the free online resources to go the extra mile if they so desire. Save your Turings and Erdoses if you can't help everyone!

Lockhart is definitely a dreamer and this isn't going to change public schools. But it might change how you as a parent get involved with your children and math.

## Re:Several Proxies (Score:5, Informative)

## Re:Several Proxies (Score:4, Interesting)

## Re: (Score:3, Insightful)

Like most natural human languages (those spoken by beings who are trying to communicate with one another and who exhibit the power of judgment), English allows some variation in what elements of sentences go where. No competent English-speaker who hasn't remained retarded in their sexual development at the anal-retentive stage, or developed a weird fetish by the daily practice of putting parentheses around symbolic expressions to coerce mechanical systems into evaluating them in the preferred order, could e

## Re: (Score:3, Informative)

I understood what he was trying to say, but observed that there was potential for confusion based upon his word order.

Being that I'm a bit of a grammar nazi when I feel like it, and that it is Friday, and that we all need a quiet chuckle on Fridays, I decided to try my hand a crafting a somewhat amusing joke based upon the lack of clarity in the sum

## Single Best Fix: Introducing Discrete Mathematics (Score:3, Interesting)

0. arithmetic

1. algebra I

2. geometry

3. algebra II

4. trigonometry

5. elementary analysis (includes some probability and statistics)

6. calculus

The above mathematics sequence is typically plug-and-chug: plug some numbers into some formulas and produce a result. No thinking is required.

What is sorely needed is a course in discrete mathematics between geometry and algebra II. Discrete mathematics te

## Re:Several Proxies (Score:4, Interesting)

The Bible says Pi = 3, in the 1st Kings (7:23). That's OT, not NT.

## Re:Several Proxies (Score:4, Interesting)

## Slashdotted (Score:4, Funny)

## Cue the other subjects (Score:5, Insightful)

The problems with K-12 education go WAY BEYOND mathematics.

## Re:Cue the other subjects (Score:5, Funny)

## Re:Cue the other subjects (Score:5, Insightful)

and most of them can be traced to certain groups (*cough*fundamentalists*cough*) waging a 30 year war on public education, and people refusing to see and treat education as what it is: an investment in the future national security and economic stability of the united states.

## Re:Cue the other subjects (Score:5, Insightful)

## Re: (Score:3, Insightful)

If everyone was smart, who would work at mcdonalds?There would always be people at the bottom, no matter how educated everyone was.

Lad:Would you like to discuss quantum mechanics? My thesis was about...Me:Just get my fucking burger.Lad:sorry sir, was this to go?## Re:Cue the other subjects (Score:5, Insightful)

If everyone was smart, who would work at mcdonalds?

Smart people. Wouldn't that be awesome?

## Re: (Score:3, Funny)

But who would eat there?

## Re:Cue the other subjects (Score:5, Funny)

But who would eat there?Feature, not a bug.

## Re:Cue the other subjects (Score:5, Funny)

But bugs already DO, don't they?

## Re:Cue the other subjects (Score:5, Insightful)

If everyone was smart, who would work at mcdonalds?

Robots obviously.

## Re:Cue the other subjects (Score:5, Insightful)

What's wrong with smart people working in the service industries?

That's somewhat of a rhetorical question, but it really isn't a great thing that we have a defacto class system based on keeping some people ignorant and poor while others enjoy luxury. We assume that working in a restaurant should be a job of lesser human beings who aren't deserving of respect, and we've ensured that those jobs don't pay a livable wage.

We complain about foreigners stealing our jobs, and we complain that poor people are so filled with vice that they don't pull themselves up by the bootstraps. Meanwhile, we make sure our economy is filled with jobs that can't provide what we consider an acceptable quality of life, and we close off routes for upward mobility wherever we can.

And if smart people did work at McDonalds, their intelligence and education still wouldn't be a complete waste. They'd still probably be better citizens, run the restaurant better, and maybe get my order right every once in a while. And who knows, maybe one of them would someday revolutionize the food service industry with innovative new ideas.

## Re:Cue the other subjects (Score:4, Interesting)

If everyone were generally qualified for interesting jobs, then boring jobs would pay much, much better. And interesting jobs would pay poorly.

That already happens with academia: the salaries at research institutions are often less than those in community colleges, simply because the former are more interesting jobs.

## Re: (Score:3, Interesting)

Is this actually an argument for the promotion of ignorance? No, it's not. It's a way for us to confirm our belief in the American Meritocracy.

Idon't w## Re: (Score:3, Insightful)

I would point out that the vast majority of senior McDonald's executives started out in a restaurant.The ratio of McDonald's executives to Mcdonald's grill / fry / sandwich / cashier persons is pretty daunting. Ultimately only a handful can rise to executive level. Even if everyone wanted to and was capable of the job their isn't room for everyone to advance.

## Re: (Score:3, Interesting)

Personally, I put the blame less on fundamentalists and more on decreasing importance of education in the home.

There are dozens of examples (single mothers with multiple jobs and multiple kids who just don't have time to parent, illegal immigrants raising kids that accept no-skill jobs as manual labor as sufficient for a lifetime instead of working to get an education and work in a skilled field), but the basic problem is that kids don't believe that they need a real education to live.

## Re: (Score:3, Insightful)

However, I think it's unfair to specifically single out these groups without including what seems to me to be a "lax" attitude by parents (including college educated parents) in middle-class homes where the kid's great-grandparents (or earlier) lived in the US (having immigrated or having been born in the US). In the public middle-class schools these kids go to, parents complain that poor Jason just doesn't have enough time after school for all his activities and his life is so stressful so the

## Re:Cue the other subjects (Score:4, Interesting)

American education was designed to fail. Read the book (it's free online) The Underground History of American Education by John Taylor Gatto. He is a former New York State and New York City Teacher of the Year

http://www.johntaylorgatto.com/underground/

## Re:Cue the other subjects (Score:5, Informative)

Before you troll and bash "fundamentalists" with no proof you should read a few books on why education in the US is in the state we now see.

The Deliberate Dumbing Down of America [deliberate...ngdown.com] By Charlotte Iserbyt

An Underground History of Education [johntaylorgatto.com] by John Gatto

Or read the Dodd Report [scribd.com] to the Reece Committee which investigated Tax Free Foundations in the early 1950's.

## Depends on what you mean by fundamentalists (Score:3, Informative)

and most of them can be traced to certain groups (*cough*fundamentalists*cough*) waging a 30 year war on public educationDepends on what you mean by fundamentalists. Honestly, I have my doubts you can trace all our problems back to creationists and prudes. You'd have to get the market fundamentalists, the "one curriculum to bind them all" fundamentalists, the Fabians, the Rothschilds, the Rockafellers, and probably more in there to get a really good idea of why we've ended up so mixed up.

That said: I got a

## Re: (Score:3, Interesting)

Of course, that group has a higher percentage than average of home-schoolers.

And those home-schoolers tend to get much more out of their education than average.

But go ahead with your beliefs.

## Re: (Score:3, Interesting)

Much earlier. Tennessee v Scopes.

While there was a moral victory for science educators at showing the issues in trying to restrict science education, most discussions I've read of the outcome of the trial point out that the end result was removing from textbooks significant material that was considered offensive.

This changed dramatically in 1957 with Sputnik. There was a brief rush to teach more advanced technical subjects. Beyond that date, I have anecdotes rather than more solid information about the s

## Re: (Score:3, Interesting)

## Re: (Score:3, Interesting)

By teaching how, rather than why.

"Here is how to find some property of a right triangle" rather than "Here are the qualities of a right triangle. What can you find using that, and why?"

While that method is useful in learning how to apply some given formula, it is useless in learning how to derive a formula or understand which one to use and why. Modern US algebra students might be able to tell you the square footage of pen they can construct with a given length of fence. Very few would be able to reverse th

## Re:Cue the other subjects (Score:4, Insightful)

The problems with K-12 education go WAY BEYOND mathematics.Amen to this.

I'd say the majority of the issues, though, start at home.

Too many families are stuck running a two-income home (for a variety of reasons) and simply can't/won't/don't spend the time needed with their children in the formative years.

A lot of the rest, IMO, can be traced to schools not teaching children how to think critically, just to memorize stuff.

And that sucks.

## Re:Cue the other subjects (Score:5, Insightful)

Even worse is the move away from competitiveness in many areas. I was a teacher for a while, and much of my teacher training was tainted by what was mislabeled "child-centered education" - basically don't do anything that might hurt the feelings of the most sensitive child you could imagine. Don't use a red pen to mark their work because that's an angry color; don't correct their spelling because that stifles their creativity; don't hold academic competitions because the kids who don't win (don't dare call them losers!) will be upset.

This trend continued despite the fact that high schools started graduating functionally illiterate and innumerate kids, even though they had passed the courses that should have given them reasonable skills in those subjects. Colleges and universities expended their gradual entry programs (basically high school subjects aimed at those who came from a disadvantaged background) until first-year studies were assumed to be nothing more than a high school refresher.

I left teaching mainly because the schools where I taught were basically big-kid daycare centers where there was very little learning to interfere with the political agendas of the administration and the school boards, but not before I subversively gave a few students the motivation to question what they were taught and learn on their own.

## Re: (Score:3, Funny)

## Re: (Score:3, Insightful)

There are certainly some good teachers that stick it out and I don't blame anyone for bailing out of an untenable situation, but society has got to recognize that education is important, equality should be about

opportunitynotoutcome, working hard is an important component to success, and teachers should be accountable (andnot"entitled" to their job just becau## Re:Cue the other subjects (Score:4, Insightful)

What studies I've seen show approximately no difference between children who have been in day care and children with a stay-at-home parent. (Okay, there's some detail differences early on, but they fade).

The important thing is the parents' attitude. Young children will emulate their parents, and will try to please them. If the parents make it clear that education will please them, and put enough time and effort into monitoring it to make that perfectly clear, and to be able to tell the difference between learning things and getting good grades, the children will respond appropriately.

A parent who wants to encourage education, and isn't totally swamped with other things that he or she is basically incapable of parenting, can find a way to do so.

## Re:Cue the other subjects (Score:5, Informative)

## I Sympathize With Him But Too Idyllic (Score:4, Insightful)

From the blog:

I defy you to read and find a single sentence that isn't permeated, suffused, soaked, and encrusted with truth.

Very well, here is an excerpt from the PDF:

Mathematics is an art, and art should be taught by working artists, or if not, at least by people who appreciate the art form and can recognize it when they see it. It is not necessary that you learn music from a professional composer, but would you want yourself or your child to be taught by someone who doesn't even play an instrument, and has never listened to a piece of music in their lives? Would you accept as an art teacher someone who has never picked up a pencil or stepped foot in a museum? Why is it that we accept math teachers who have never produced an original piece of mathematics, know nothing of the history and philosophy of the subject, nothing about recent developments, nothing in fact beyond what they are expected to present to their unfortunate students? What kind of a teacher is that? How can someone teach something that they themselves don't do? I can't dance, and consequently I would never presume to think that I could teach a dance class (I could try, but it wouldn't be pretty). The difference is I know I can't dance. I don't have anyone telling me I'm good at dancing just because I know a bunch of dance words.

Now I'm not saying that math teachers need to be professional mathematicians--far from it. But shouldn't they at least understand what mathematics is, be good at it, and enjoy doing it?

Well if you're not asking for teachers needing to be professional published mathematicians, what was that paragraph about?

... everywhere. That art teacher that actually made you think about what 'art' is? Not going to find many of them in the political science department, are you? Of course, for any subject, someone who puts their heart and soul into the subject is the best teacher! In this respect, math is not special.

I'm sorry man, you're asking for the perfect math teacher. You know Robin William's character from the movie

The Dead Poet's Society? You want a guy like that for mathThe paragraph I quote is not the truth, it's wishing for the impossible. I wish I had a math teacher like this all my life but come on. The public school system is more worried about getting someone that actualy cares about the students at all. They can't even find those people let alone people who care about the students

andlive/eat/sleep/bleed math.I'm right their with you in wishing for this but the expectation is unrealistic. Passions come to people unexpectedly. We should deal with the fact that more people are passionate about topics like Art and Humanities than Math and Computer Science. It's just the reality of academia right now.

## Re:I Sympathize With Him But Too Idyllic (Score:4, Insightful)

you don't have to be a PhD. to be interested in and passionate about math. there are some very elegant things in math, and if they are taught to kids in the spirit of a voyage of discovery rather than a trudge along the banks of the river Styx, then there's a chance more kids will catch the bug and like math. And at the rate we're losing engineering capability, particularly in the US, this ought to be a priority.

## Re:I Sympathize With Him But Too Idyllic (Score:4, Interesting)

Passions come to people unexpectedly. We should deal with the fact that more people are passionate about topics like Art and Humanities than Math and Computer Science. It's just the reality of academia right now.

Isn't his point that we don't really know if that's true, since math isn't taught in a way to inspire passion? That if more people were able to glimpse some of the beauty and creativity in it, there might be more interest in it?

Well if you're not asking for teachers needing to be professional published mathematicians, what was that paragraph about?

I agree we can't expect every teacher to be awe-inspiring; even getting (and retaining) enough marginally competent teachers is a challenge. However, you needn't be a university-level mathematics professor to know some of what he's suggested. For example, public school teachers are supposed to have Master's degrees, right? Now, isn't there something funny about the fact that teachers will go and get their BS in the subject they will teach, but get their Master's degree in "education"? Cue the "Philosophy of science is about as useful to scientists as ornithology is to birds" quotes. I'd think that teachers might be better served by a decent master's degree in their field of teaching, rather than "education". That would allow them the opportunity to study the history and philosophy of their subject, get a grasp of recent developments (maybe not in all subjects, but they could at least be able to pick up journals), etc. The really good ones could even get published (I just got my Master's degree, and was able to get a paper published, so yes, it's possible).

## Re: (Score:3, Informative)

## Re:I Sympathize With Him But Too Idyllic (Score:5, Insightful)

To be honest, I thoroughly disagree with you, because I

DID HAVEjust such a teacher. She wasn't some kind of superwoman either, she was just very competent at math (no advanced degrees, but good enough to teach basic calculus, algebra, and geometry in a way that made pretty much all the students at my school respect her). More importantly, she was passionate about giving students a fundamental understanding of the subject matter. She didn't want to just cross her T's and dot her I's and be done with it, she wanted us to learn what it was all about. She was a hard teacher, but she was almost remarkable in that nearly the entire student body had a great deal of respect for her.I think the author's whole POINT was that it's claims like yours--that this is some kind of unreasonable expectation--that are entirely the problem with the situation we have. The simple fact is,

it is not unreasonable. My personal experience has shown me that there ARE such teachers out there; mine as well as others I've known.My own personal take is that our society simply doesn't give educators the respect they deserve. There's very little motivation for the kind of intelligent, competent, passionate people to go into to lower tiers of the world of education. We pay them peanuts and there's not nearly the kind of appreciation and respect out there for them to want to do those jobs. I happened to go to a private Catholic school, where neither of those things are true, and let me tell you the difference was obvious.

## Re: (Score:3, Interesting)

The public school system is more worried about getting someone that actualy cares about the students at all.Now who's being the idealist? The public school system rarely has such concerns, or they wouldn't do everything possible to scare away the best teachers, and even moderately good ones. Standardized testing, nonsensical state mandates, psychotic district administrators, requirements to use ghastly textbooks, etc. So many headaches are thrust upon our public teachers that have nothing to do with

## Re:I Sympathize With Him But Too Idyllic (Score:4, Insightful)

We should deal with the fact that more people are passionate about topics like Art and Humanities than Math and Computer Science. It's just the reality of academia right now.

Of course it is, because we have these ridiculous stigmas:

Artis passionate, frivolous, and beautiful.Mathis boring, uninspiring, and useful.What?! There is no such thing as frivolous beauty; no utility is uninspiring and cold. Lockhart, I fear, misses this point. I understand the frustration Lockhart feels at the 'math = boring' stigma, but countering that 'math = art' is just as damning in our obsessed-with-mutual-exclusion society.

Beauty and utility have long been a happy couple. The false rumors of their divorce is, I think, the root of Lockhart's (and my) frustration.

## Re: (Score:3, Insightful)

The only way this guy would get the class he wants is to only teach elective courses that aren't pre-reqs for

## Re: (Score:3, Insightful)

It's not necessary to put a perfect math teacher in every classroom. Elementary school teachers are perfectly capable of teaching a math curriculum that presents kids with mathematical concepts in game form as Lockhart mentions. Later in their school careers the kids who show interest and aptitude for math could be taught algebra etc, and the rest could stick with the mechanics of arithmetic that will enable them to deal with checkbooks and mortgages. I think our problem today is that we use a one-size-fits

## Re: (Score:3, Insightful)

In history, a lot of very prominent mathematicians of their day and age made their living privately teaching high school kids. In modern times, mathematics isn't seen as an important an asset to have to spend that kind of money even if one has it.

I don't understand why education is seen the way it is in the US. What does the teacher have to do with the quality of education? Does anyone

## it's really bad (Score:5, Insightful)

High school students are forced to write proofs as part of geometry class. However, they are never taught the rules of logic before being asked to write these proofs. That is just one example of how horribly, horribly stupid the HS math curriculum is in the US.

## Re:it's really bad (Score:5, Insightful)

I agree, but that's not my point. My point is that understanding the formal rules of logic is fundamental to being able to understand proofs. But the bureaucrats who came up with the US math curriculum just said "the kids should learn this and this and this" but never attempted to put those things in the right order so that it was even possible for them to learn all those things.

It's no wonder kids think they are bad at math or hate the subject--it is presented to them in an impossible form.

## Re:School Choice is part of the answer (Score:4, Insightful)

1) Yes NCLB is an excuse to close down public schools - it was designed as such, and they intentionally fabricated a study [exposed as a fraud 2 years later] to get congressional support

2) Defunding them further with vouchers (most of which would be going to religious indoctrination centers that masquarade as schools) is not a solution

3) "Failing schools based on geopgrahy" is a problem with two things

a) How we fund schools [how about pool money state wide and dole out as needed instead of tying funding to their service areas land values.. that kidn of funding arraignment was obviously designed to serve only the rich neighbors]

b) home lives in disadvantaged areas are more often than not are harmful to getting an education.

The NEA would CORRECTLY resist #2. They would support repleaing NCLB and getting all schools funded better.

Vouchers are not a solution, they're just a furtherance of stripping funding from public schools so that they fail.

## Re: (Score:3, Informative)

## The way math is structured is disconnected from... (Score:4, Interesting)

... interesting things kids want to do.

Lets face it a minority of people will like math, but matehmaticians have done a lot to make mathematics overly complicated.

I struggled with the symbolic format math was presented in highschool because it was so disconnected from the world, only as I got older did I realize how arbitrary and how that was only one way to present mathematics. To really teach math one must learn how to observe first before one even gets into symbolic computation, math at it's most basic is about observing relationships, patterns of : Size, ratio, proportion, etc. It's really a language invented to systematize structure and relationships of the real world, therefore how math is represented and structured and is taught matters a hell of a lot.

I've learned over the years that many mathematical systems are totally arbitrary are are more obtuse then they need to be, math comes from the simplest observations. Math has built up a lot of cruft and wasteful jargon disconnecting math from the world.

For instance I had no idea for a long time that the way math is structured could be restructured when I was young and it was one group of peoples perspective on mathematical principles, I came across debates and alernative systems like:

http://www.symmetryperfect.com/ [symmetryperfect.com]

And it showed me how arbitrary mathematical systems and their structures really are and they are built to suit particular kinds of minds or cultures.

For instance the ancient mayans used shapes for numbers, instead of 1, 2, 3

See here:

http://en.wikipedia.org/wiki/Maya_numerals [wikipedia.org]

Math is a very rich subject which unfortunately has a lot of cultish like people who think themselves the gatekeepers of mathematics.

I've thought about writing a book in my spare time about how badly mathematicians and the academia has blinded themselves to simplifying mathematics by focusing too much on symbolic jargon and not teaching children how 'mathematical' relationships are related to our simplest observations of the world: Size, shape, form, color, motion, etc.

## Re:The way math is structured is disconnected from (Score:5, Informative)

For instance the ancient mayans used shapes for numbers, instead of 1, 2, 3

Psst! The numerals "1", "2", and "3" are shapes too!

F***in' indocentrists...## Re: (Score:3, Insightful)

so you want to teach math using base-1 ... that's... insane.

## Re: (Score:3, Insightful)

What your holding as a genius way to do things is no different then Roman numerals. It gets extremely unwieldily for anything other then simple addition and subtraction or basic counting, and they aren't any less arbitrary in their symbolism then Arabic numerals are.

## Re: (Score:3, Interesting)

If you read the Wikipedia article on Maya numerals [wikipedia.org], linked to above [slashdot.org], you will see that it is

notlike Roman numerals. It is, in fact, a base-twenty positional system that happens to have logical symbols for its digits (zero notwithstanding).## Re:The way math is structured is disconnected from (Score:5, Insightful)

## Re: (Score:3, Insightful)

No. There's always some college that will take you, even if you got average grades (and below average, people probably aren't interested in college anyway). Sure, you might not get a scholarship and have to take out burdensome student loans, but when American culture now emphasizes that a college degree is for everyone, and universities are businesses after your money, it's a buyer's market.

## Re: (Score:3, Insightful)

"Dude, now you're approaching xenophobia. Have you looked at the state of mathematics in American universities? A conspicuous amount of highly original researchers are the product of foreign educational systems. They aren't doomed to being tech support monkeys like you insinuate."

I don't know how the heck you got that out of my statement. I'm not saying these people are not smart, I'm not saying they are not mathematical prodigies, but they have all learned math in a particular way, many mathematicians don

## Re: (Score:3, Insightful)

A lot of math reads like extremely bad Perl programs too, with tons of functionality on every line and no documentation except for a giant paragraph at the top written by someone who is apparently from Mars.

On the other hand, a lot of math is just pattern recogni

## True story .... (Score:5, Insightful)

In university, I was taking an intro philosophy course on critical reasoning.

We had covered the concept of statistical significance. The example we'd used was a case of "0.05" meaning we had 95% confidence in the statistical results. On the exam, the professor made a typo, and the question read "how much certainty with a statistical confidence of 0.5", to which the correct answer is 50%.

I was marked as wrong, and when I complained, the professor indicated that since we'd never covered that example, and only covered 0.05 in class, it was assumed that was what she meant.

I informed her for someone teaching critical reasoning, she wasn't demonstrating any. I also insisted I get the credit for giving the actual correct answer (which I and everyone who answered it correctly did).

If that's indicative of how math is taught nowadays, we're all hosed. :-P

Cheers

## Eh. (Score:5, Interesting)

The whole idea behind his essay is that he liked playing with numbers and shapes as if it's an art, but he doesn't seem to realize most people don't share this love for math, like pretty much 90% of any student population. This is me speaking as a just-graduated senior: the things he suggests is beyond the ability of most math students in high school.

## Re: (Score:3, Informative)

The whole idea behind his essay is that he liked playing with numbers and shapes as if it's an art, but he doesn't seem to realize most people don't share this love for math, like pretty much 90% of any student population. This is me speaking as a just-graduated senior: the things he suggests is beyond the ability of most math students in high school.

I think you missed the point.

His point IMO, is what we are teaching as "math" in school is totally useless and should be scrapped completely, because it's not even close to what math is.

We don't need to teach math to 100% of the students, just as we don't insist that 100% of the students can paint landscapes, or bake brownies.

## Re:Eh. (Score:4, Insightful)

I read as much of the essay as I could before I realized that the guy doesn't understand that his experience doesn't apply to everyone else. I understand where he's coming from because I tell the worst stories imaginable. I will go on talking about little, highly interesting details, until I realize that I'm the only one who finds them interesting. It took me a long time to realize that, just because I find it interesting, that doesn't mean that other people will.

To say that mathematics should be taught in the way that he likes the most is silly, at best. Most people will be able to pass through life with a rudimentary, at best, understanding of mathematics. Most jobs in this world do not require 90% of the theorems and principles that people are forced to learn through high school. I agree with the essay 100% on that point.

The key to math education, though, is not memorizing these principles, but rather learning how to solve problems. If someone can logically plan their way through a calculus problem, almost anything that they have to figure out at their job would be well within reason.

I never have understood the concept of math as an art, yet I enjoy math. I enjoy solving problems, enough so that I earned my BS in Mathematics, but this guy takes it to a whole new level. If not even all mathematicians think like he does, why does he expect that the general population will?

## Re: (Score:3, Insightful)

90% of the student population isn't interested in math because it's taught in the way and by the people he talks about in his text.

*insert snide comment about reading comprehension here*## Re: (Score:3, Insightful)

You overlooked the absolute core of his argument: a lot of high school students think they don't like math because they've been presented with a pale shadow of math for the previous eight years. Of course a high school student couldn't handle what he's describing; all of their previous schooling has emphasized rote memorization, blind pattern matching, and robotic application of rules. He thinks we need to rethink things all the way back to first grade.

By odd coincidence, I had an english literature tea

## Re: (Score:3, Informative)

Is it the students who are incapable, or are they merely inexperienced?

Some people just don't get math, just like some people just don't get music or science or English. Around here, the math one is probably the largest.

You can ask any teacher. Some students just have a hard limit on their abilities. It's hard to manage these.

I'd also like to state that we have some pretty damn good teachers in our district, who approach math more like the essay stated. Lots of exploration and discovery. For a lot of students it just doesn't work, whether they're told outright (which the ma

## Hmm (Score:5, Funny)

I implore you to read his essay with every atom of my being.

Well, OK, seeing as I can use *your* atoms.

## Oh give it a rest (Score:5, Insightful)

Instructors of just about every field at any level of compulsory education (K-12) have to battle against entrenched biases against their fields, and against education in general, that have been fostered for years before the student ever gets in their classroom. Further, their task is to teach the curriculum provided. If they inspire their kids to love the field, that's great, but if they spend so much time inspiring the kids that they don't have enough time to teach the kids what they need to pass the state-required tests, they're still going to lose their jobs.

Teaching math, science, or anything else is HARD. Teaching it to people who don't care and don't want to be there is even harder. Teaching kids to love the field when the only metric used to judge your performance is pass rates on a standardized test is harder still. It's all well and good for professional mathematicians to bitch and moan about the state of education, but until they're ready to step in with some realistic and implementable ideas that don't presuppose that all kids have some inherent interest in these things that just needs to be tapped into, it's not helpful in the least.

## Re:Oh give it a rest (Score:4, Insightful)

Specialists in every field complain that educators get their field wrong or don't stir the passions of kids for their field as much as they ought to. What they fail to understand is that they're coming at the whole problem from the perspective of someone who is obviously gifted at and highly passionate about the field. They don't seem to get that most people don't pick up their field as easily as they do, and don't care enough to put in the effort it would take to get even half as good at it as the specialist.

Do musicians complain that the typical high school band teachers don't understand the basics of music? This is a specific example from the TFA and it is very well chosen. People don't expect high school band teacher to world class musicians. They do however expect high school band teacher to have a feel for what music is. They expect high school band teacher to know the difference between in tune and out of tune. They expect high school band teachers to drill notation and teach counting different times, but the also expect to be connecting these things to actual music at every step of the way.

We expect this of high school band teacher because most people know what music is supposed to sound like. Most people have enough sense for how it actually works to recognize somebody who can't play, or who cannot teach how to play.

Teaching math, science, or anything else is HARD. Teaching it to people who don't care and don't want to be there is even harder. Teaching kids to love the field when the only metric used to judge your performance is pass rates on a standardized test is harder still. It's all well and good for professional mathematicians to bitch and moan about the state of education, but until they're ready to step in with some realistic and implementable ideas that don't presuppose that all kids have some inherent interest in these things that just needs to be tapped into, it's not helpful in the least.

If you tried to teach a music class based on transcribing notation and chord theory, rather than listening and/or playing you'd find it hard also. Teaching kids to love music using a such a curriculum wouldn't just be hard, it would border on the absurd. Even if a few people did enjoy the raw mindless diligence to do such a thing out of context, there is no particular reason to believe that this would produce great musicians.

I'd like to add that science education in the US seems to me to be much closer to math education than music education. I remember learning to play lip service to the scientific method, but I don't remember ever being asked to sit down with some lab equipment and figure out what some relationship is. If you are given the equation, and given the experiment to "test" some particular aspect of the equation, you've removed the science, you've removed what is important.

## An allegedly true story from a professor (Score:5, Interesting)

While I was in university, a computer science professor in the faculty of mathematics told me (and the rest of the class) a cute and funny story about what happens "when the children of math professors get together". He and a colleague, who each had a young daughter at that time, were walking together in a park with their daughters. The children were old enough to have picked up some math-related words and phrases from their fathers, but young enough to have no idea what they really meant - six or seven years old, maybe? The daughters went off to play and their fathers overheard them arguing about who had seen the most flowers in the park.

My professor's daughter said, "I saw five flowers!"

"And I saw... six!", the other girl replied.

Not to be outdone, my professor's daughter said, "I saw a million flowers."

"Oh yeah? I saw infinity flowers."

This, according to my professor, caused his daughter to pause - she had never heard of "infinity" before. How could she top "infinity flowers", especially since she didn't know what it meant?

But after thinking for a few seconds, she said, "Well, I saw

allthe flowers."## US School System compared to Europes School System (Score:5, Informative)

I myself have gone through the US school system starting at grade 7 (lived in Switzerland and The Netherlands before then), I am currently in uni for a software engineering degree. While I have read only part of the article (the blog post) I wanted to post my experience compared to that of my cousin who went through school in The Netherlands.

Math at the schools I went to was catered to the lowest common denominator, the slowest person in the class, the person who would just not get it got the most attention and the rest of the class was stuck at that level until that person tagged along and finally got moving. Whereas in Europe and other places they place those students in various levels of math dependent on their skill level so that those that don't need the extra time are able to get to the higher level maths faster. This creates a gap between the math that is considered required at age 18 in the US and The Netherlands. My cousin was going for a degree in hotel management and food preparation (chef). He at the age of 18 had a better understanding of math, and had more knowledge of high level math (Linear Algebra, Calculus and others) than I did when I graduated High School, and the classes he were in were considered the slower less demanding classes since it was not as much of a requirement for the degree he was going to be pursuing.

This is the same with a lot of the classes though, history, english, and science classes. Especially for English, you don't get to think for yourself anymore, you have to follow exactly what the teacher told you. If the teacher says this is important for this reason, and you attempt to argue it differently in a paper you fail, everyone coming out of high school has been passed through a cookie cutter, there is no innovation left, there is no real thinking for oneself anymore.

It is sad, and the state the US educational system is currently in will not allow it to compete in the global market, it will not allow it to be innovate and provide new ideas, but what it will provide is people who are like sheep and are more than willing to follow the crowd and just do it because everyone does. These people will be easy to govern and control since they won't ask questions and least of all will they rebel and fight for their beliefs. In other words, the US education system as it currently stands is making zombies.

## Re:US School System compared to Europes School Sys (Score:4, Insightful)

You must have attended a very very small school. Most US schools have different courses based on skill level. Your conclusions about the US school system are therefore wrong. They are merely conclusions about very small schools.

## Re: (Score:3, Interesting)

I saw the issue the OP stated in all of the K-12 schools I went to. As my father was in the military, I got to go to 3 elementary schools, 1 middle school, and 2 high schools. Some teachers were able to handle students at different reading/math levels in elementary school, but once I hit middle/high school, everything except math was lowest common denominator. In Seventh grade, the English class was using the reader I used in Fifth grade. And people in the class were having a hard time with it! The onl

## Re: (Score:3, Informative)

## You can convince me (Score:5, Insightful)

that math is better taught as an art than as a pragmatic problem-solving toolset when you can convince me that Pablo Picasso should have been forced to paint the Golden Gate bridge.

Society needs math as a tool in far greater quantity than math as an art. Socially-funded education serves the greater need of society. QED.

I survived public school mathematics. I still appreciate the beauty of patterns, especially the relatedness of art, music, and math. (

Godel, Escher, and Bachreally resonated for me. But that didn't make me a mathematical artist, any more than a musical composer or a woodblock printer.)Lockhart's essay is an interesting read, really, but on some level it boils down to "Those unworthy schlubs treating Mathematics as a tool don't deserve it. It belongs to the artists, the dreamers, the purists!"

It's a pretty common arrogation in the math culture, it seems. I dont' recall sculptors ever being pissed at concrete workers or ironworkers. And I've never heard of any artist painter getting mad at the other kind of painter for not employing good artistic composition principle while painting the side of the barn.

Seriously. Math is both an art and a tool. The best artists find their art by themselves; they're not turned out by artist factories. School mathematics is to turn out the mathematical equivalent of bridge painters and ironworkers, because society needs those more (in greater quantity).

## Re: (Score:3, Insightful)

Part of his argument is that by focusing on "math as tool" via "math as rote memorization", you fail even at that. Math at higher levels becomes cryptic symbols that you manipulate according to cryptic rules to make your teacher happy. And a few years out of school you promptly forget the whole thing. If they forget most of it, and for the overwhelming majority it never hurts them to have forgotten it, what was the point in having "taught" it in the first place? He argues that it's such a waste of time

## Two mathematicians (Score:4, Funny)

Two mathematics professors are having lunch at a restaurant. The first mathematician keeps complaining about how ignorant the typical American is and how he's suprised that the average person in this country has enough mathematical prowess to balance a checkbook.

The second mathematician says, "Don't you think you're being a little harsh? The average person surely has more mathematical ability than you give them credit for."

The first mathematician responds, "Absolutely not! I'm sure if you asked the first person you met on the street to solve a basic algebra problem, they would have no idea where to start."

The second mathematician says, "Okay, I'll make a bet with you. At the end of the meal, I'll ask our waitress to solve a calculus problem. If she can solve it, you pay for lunch. If she can't, I'll pay."

"Thanks in advance for lunch!" the first mathematician says confidently.

Later, while the first mathematician is in the bathroom, the second mathematician flags the waitress down and says, "Listen, when you bring us our check I'm going to ask you a math question. I want you to answer, âone-half x-squared.' Can you remember that? If you do, I'll leave an extra big tip." He encourages her to write it down phonetically and practice it so that it seems natural.

At the end of the meal, after the waitress puts the bill on the table, the second mathematician says, "Oh, could you answer a little question for me? What's the integral of x with respect to x?"

The waitress looks unsure at first, but says, "One-half x-squared."

With a grin, the second mathematician slides the bill over to the first mathematician.

As the waitress is walking away, she turns back and says over her shoulder "plus a constant!"

## Half Steps (Score:4, Insightful)

First, *use* math to solve real problems and explain real scientific principles. Radio Lab (THE official National Public Radio show for geeks everywhere) had a great little episode where some student "discovers" that the periodicity of a pendulum forms a parabola when charted on a graph. Wow! That's heady stuff. (It's the first story of this episode [wnyc.org].) Understanding the interaction of science and math -- the universe, really -- is something that we can teach. Integration of math and science gets us part of the way there.

Second, incorporate the history of math into math class. Math advances all occur because of some historical context. Combining the two is a half-step that will get students to understand "why" we created this math, even if they never quite get the quadratic formula down. Combine these two principles, and it would go a long way.

## U.S. Public Education (Score:3, Interesting)

Discussing "US Public Education" is about as specific as discussing global weather. Is it cloudy or raining today? The education system is the US is quite federalized- most of the decisions about pretty much anything are made at the state and local levels.

I, personally, am quite happy with my 1st graders' (twins) math education. They've learned concepts like how to estimate, pattern detection, etc., as well as the rote mechanics of arithmetic. And they get more of it at home ("Here's a cookie. Tomorrow I'll give you twice as many as I did today. How many will you have in a week?"). But I live in a pretty rich suburb outside Boston, where the MIT professors live in the less-affluent neighborhoods.

We can bitch about the schools all we want, but it's a deeper cultural issues. School teachers get OK pay and benefits, good (though rigidly defined) vacations, and no respect. What kind of profile of person does that attract? In my experience, a real mix of people who are passionate about teaching (often with well-paid spouses) and those that mail it in 'til vacation starts. The balance of those (and other) groups varies widely by district. More than pay, this is really an issue of respect. I can't tell you how many teachers I know who report 'lack of respect for their profession' as the #1 gripe about their job. I wouldn't put up with that (not that I'd make a good teacher).

## More money in death (Score:4, Insightful)

I have to comply with 300 pages of regulations for the school I started in Denver. The cost of compliance is at least half the total budget.

Although this article did not touch once upon the issue of wages, it is a very good article -- perhaps the best I've read all year on the subject of education. The need to introduce mathematical intuition at a young age is something the Montessori Method has done for a century. In a Montessori school, the child progresses from concrete to abstract, working first -- from very young at two years old -- with physical objects that embody length, area, or volume, and only later attaching the abstract symbols we call numbers. The physical manipulation leads to visualization of how addition, subtraction, multiplication, division, and fractions work. A child who goes through all three years of "Primary", which is age 3 to age 6, by the end of it, the child will be multiplying and dividing, and have worked with manipulative materials that demonstrate fractions and even binomials and trinomials from algebra.

In the face of competition from government schools, it is a challenge. I have learned that the competition isn't so much for students as it is for teachers. By using tax dollars, they can pay so much more, offer more benefits, and provide stability stemming from a legally-guaranteed funding sources. Meanwhile, the government schools are there for the purpose of creating cannon fodder, with its flag worship every morning and the forced admission of military recruiters under No Child Left Behind for as early as third grade. And when they do grab a hold of an effective pedagogy like Montessori, they pervert it by adding standardized testing and segregating by ages (e.g. two-year age groups rather than the three-year age groups prescribed by Montessori).

By eliminating public education, and by reducing the morass of regulations for running a private school, the free market could decide how important math education really is, rather than hearing hot air about it from public officials and CEOs, or by listening to earnest mathematicians such as Paul Lockhart, the author of this white paper, attempt to influence curriculum, presumably in government schools. The century-long battle between phonetics and "whole word" in the area of language (and the resulting reading levels no matter what is done) should be evidence enough of the futility of this approach (to use an anlogy, which Lockhart seems to love).

## A teachers take (Score:5, Insightful)

Would our culture demand effort from our students instead of recognizing talent we'd be much further along.

I'm not suggesting that talent should go un-nurtured but, at least from an educators point of view, the effort of the students should be the focus of rewards.

## Housecats (Score:3, Insightful)

The author wants to a priori assume that everyone will love math if only the beauty of it is shown to them. This is mistaken. There are many beautiful things in the world that lots of people have absolutely no interest in. If you tried to force the beauty on everyone then you'd cheapen the entire meaning of beauty. Yes, in the lower grades math is a pretty mechanical thing that just equips human beings with the basics but that is OK. The Einsteins take to that like a fish to water and do their amazing work because they see the beauty without anyone saying "HEY, appreciate!"

## Speaking of hot mathematicians (Score:5, Informative)

http://www.danicamckellar.com/ [danicamckellar.com]

I can't believe Summer Glau is the chick geeks are hot after. Danica is Hot, has her name on a physic theorem, mathematician, and has written math books for girls.

Her acting career is full of geek as well.

Not to say either one of them is a geek, just that I scratch my head over why geeks prefer Summer.

## Mathematician, Taught Public HS for 5 Years (Score:3, Insightful)

Though I don't need the rhetoric, this hits it on the head, in every aspect.

I'd like to try teaching math like English -- Math 1, Math 2, Math 3, Math 4, with curriculum determined in part by such apparently meaningless factors as what might be useful in other classes or what's happening, you know, outside of my room.

The textbook comments are particularly right on -- step 1, burn them. If teachers complain that they won't know what to teach, fire them on the spot.

Geometry is also a lousy place for proof. Teach deduction all the time, in every topic -- and in classrooms other than math. "Here's a bunch of fake stuff you don't know anything about that's hard to draw. Now let's think really abstractly about how we're thinking about it!" And induction doesn't get taught at all.

The practical deal-killer, the one that drove me out of the profession, is that the barrel full of math teachers is so close to empty that you're pretty much scraping bottom from day 1. This kind of instruction -- and this kind of critique -- can only originate with someone who likes math, and is sort of good at it. You'd be amazed (or maybe you wouldn't) at how few public high school math teachers this describes.

America has gotten the math teaching instruction it asked for when it decided to prop up bad teachers with lousy but easy-to-use texts, and to boot it got the benefit of not having to pay very well for people willing to go through these motions. (It's not about money, but really, it's a little bit about money. I doubled my salary when I left last year.) It's a big, huge problem, and since you're going to have to convince parents that it needs the kind of dramatic overhaul this (great) article describes, and since parents were largely victimized by the existing system, I'm pretty sure it's a losing battle.

## This is now a book (Score:3, Informative)

## Re:tl;dr (Score:5, Interesting)

Bingo, and that's one of the big problems with trying to do anything about the issues the paper raises: there are only so many people with the 1) ability, 2) knowledge, and 3) inclination, to do the kind of

realmathematics he's talking about.We'd have to re-vamp our teacher training along the lines of what's talked about in the paper to try to increase the number of people who could do it, and

hopeLockhart's right about this being an art with universal appeal so that enough of the teacher candidates "get" it. Even if elementary schools began using dedicated math teachers (some already do, but many don't) we'd still need a shitload of people trained in this "math as an art/math as play" style, and we currently have approximately zero in elementary education.## Re: (Score:3, Interesting)

Bingo, and that's one of the big problems with trying to do anything about the issues the paper raises: there are only so many people with the 1) ability, 2) knowledge, and 3) inclination, to do the kind of

realmathematics he's talking about.And not just the teacher training. This goes beyond what some students are capable of and can handle. What happens to them if they can't function inside a creative mathematical atmosphere?

## There are _lots_ of people (Score:3, Insightful)

... with the ability, knowledge and inclination. The real problem is that they can all make twice or more money by doing some other line of work. This is a matter of paying what is necessary to compete with the other possibilities open to mathematically able, knowledgeable and inclined people.

## Re:It starts with the textbooks. . . (Score:5, Interesting)

And you get around the economic obstacles by subverting the system: Crowdsource the textbook to a group of interested mathematicians. Publish it online for free, with printed copies available for a price far below what a crooked textbook publisher would charge. Add value by posting demonstrations by mathematicians, math historians, and math professors on YouTube, linked to the relevant chapter of this comprehensive, global mathematics resource.

## Re: (Score:3, Insightful)

Wow, you didn't read the paper very closely if you think he could produce a textbook (or a series) to implement what he's advocating.

## Re: (Score:3, Insightful)

## Re: (Score:3, Insightful)

While I appreciate where you're going with your statement: "the way the discoverer came to und